Calculus Examples

$f (x) = 3 a^{3 x}$

Step 1

Find the first derivative.

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Step 1.1

Since $3$ is constant with respect to $x$ , the derivative of $3 a^{3 x}$ with respect to $x$ is $3 \frac{d}{d x} [a^{3 x}]$ .

$3 \frac{d}{d x} [a^{3 x}]$

Step 1.2

Differentiate using the Generalized Power Rule which states that $\frac{d}{d x} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = a$ and $g = 3 x$ .

$3 (\frac{d}{d x} [a] (3 x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 1.3

Differentiate.

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Step 1.3.1

Since $a$ is constant with respect to $x$ , the derivative of $a$ with respect to $x$ is $0$ .

$3 (0 (3 x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 1.3.2

Simplify the expression.

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Step 1.3.2.1

Multiply $3$ by $0$ .

$3 (0 (x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 1.3.2.2

Multiply $x$ by $0$ .

$3 (0 a^{3 x - 1} + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 1.3.2.3

Multiply $0$ by $a^{3 x - 1}$ .

$3 (0 + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 1.3.2.4

Add $0$ and $\frac{d}{d x} [3 x] (a^{3 x} \ln (a))$ .

$3 (\frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 1.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 (3 \frac{d}{d x} [x]) (a^{3 x} \ln (a))$

Step 1.3.4

Multiply $3$ by $3$ .

$9 \frac{d}{d x} [x] (a^{3 x} \ln (a))$

Step 1.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$9 \cdot 1 (a^{3 x} \ln (a))$

Step 1.3.6

Multiply $9$ by $1$ .

$f' (x) = 9 a^{3 x} \ln (a)$

Step 2

Find the second derivative.

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Step 2.1

Since $9 \ln (a)$ is constant with respect to $x$ , the derivative of $9 a^{3 x} \ln (a)$ with respect to $x$ is $9 \ln (a) \frac{d}{d x} [a^{3 x}]$ .

$9 \ln (a) \frac{d}{d x} [a^{3 x}]$

Step 2.2

Differentiate using the Generalized Power Rule which states that $\frac{d}{d x} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = a$ and $g = 3 x$ .

$9 \ln (a) (\frac{d}{d x} [a] (3 x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $a$ is constant with respect to $x$ , the derivative of $a$ with respect to $x$ is $0$ .

$9 \ln (a) (0 (3 x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Multiply $3$ by $0$ .

$9 \ln (a) (0 (x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 2.3.2.2

Multiply $x$ by $0$ .

$9 \ln (a) (0 a^{3 x - 1} + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 2.3.2.3

Multiply $0$ by $a^{3 x - 1}$ .

$9 \ln (a) (0 + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 2.3.2.4

Add $0$ and $\frac{d}{d x} [3 x] (a^{3 x} \ln (a))$ .

$9 \ln (a) (\frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 2.4

Raise $\ln (a)$ to the power of $1$ .

$9 (\frac{d}{d x} [3 x] (a^{3 x} (\ln^{1} (a) \ln (a))))$

Step 2.5

Raise $\ln (a)$ to the power of $1$ .

$9 (\frac{d}{d x} [3 x] (a^{3 x} (\ln^{1} (a) \ln^{1} (a))))$

Step 2.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 (\frac{d}{d x} [3 x] (a^{3 x} {\ln (a)}^{1 + 1}))$

Step 2.7

Add $1$ and $1$ .

$9 (\frac{d}{d x} [3 x] (a^{3 x} \ln^{2} (a)))$

Step 2.8

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$9 (3 \frac{d}{d x} [x]) (a^{3 x} \ln^{2} (a))$

Step 2.9

Multiply $3$ by $9$ .

$27 \frac{d}{d x} [x] (a^{3 x} \ln^{2} (a))$

Step 2.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$27 \cdot 1 (a^{3 x} \ln^{2} (a))$

Step 2.11

Multiply $27$ by $1$ .

$f'' (x) = 27 a^{3 x} \ln^{2} (a)$

Step 3

Find the third derivative.

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Step 3.1

Since $27 \ln^{2} (a)$ is constant with respect to $x$ , the derivative of $27 a^{3 x} \ln^{2} (a)$ with respect to $x$ is $27 \ln^{2} (a) \frac{d}{d x} [a^{3 x}]$ .

$27 \ln^{2} (a) \frac{d}{d x} [a^{3 x}]$

Step 3.2

Differentiate using the Generalized Power Rule which states that $\frac{d}{d x} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = a$ and $g = 3 x$ .

$27 \ln^{2} (a) (\frac{d}{d x} [a] (3 x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $a$ is constant with respect to $x$ , the derivative of $a$ with respect to $x$ is $0$ .

$27 \ln^{2} (a) (0 (3 x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $3$ by $0$ .

$27 \ln^{2} (a) (0 (x a^{3 x - 1}) + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 3.3.2.2

Multiply $x$ by $0$ .

$27 \ln^{2} (a) (0 a^{3 x - 1} + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 3.3.2.3

Multiply $0$ by $a^{3 x - 1}$ .

$27 \ln^{2} (a) (0 + \frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 3.3.2.4

Add $0$ and $\frac{d}{d x} [3 x] (a^{3 x} \ln (a))$ .

$27 \ln^{2} (a) (\frac{d}{d x} [3 x] (a^{3 x} \ln (a)))$

Step 3.4

Raise $\ln (a)$ to the power of $1$ .

$27 (\frac{d}{d x} [3 x] (a^{3 x} (\ln^{2} (a) \ln^{1} (a))))$

Step 3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 (\frac{d}{d x} [3 x] (a^{3 x} {\ln (a)}^{2 + 1}))$

Step 3.6

Add $2$ and $1$ .

$27 (\frac{d}{d x} [3 x] (a^{3 x} \ln^{3} (a)))$

Step 3.7

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$27 (3 \frac{d}{d x} [x]) (a^{3 x} \ln^{3} (a))$

Step 3.8

Multiply $3$ by $27$ .

$81 \frac{d}{d x} [x] (a^{3 x} \ln^{3} (a))$

Step 3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$81 \cdot 1 (a^{3 x} \ln^{3} (a))$

Step 3.10

Multiply $81$ by $1$ .

$f''' (x) = 81 a^{3 x} \ln^{3} (a)$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $81 a^{3 x} \ln^{3} (a)$ .

$81 a^{3 x} \ln^{3} (a)$